A rambling narrative of one man's journey through mathematics

Abstract Algebra

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Despite my hope to the contrary, it would appear that the math I’ve done while here so far as not parlayed into me blogging super-frequently. For what it’s worth: Life is busy. Just in case you were wondering. ^_^

Lately, I’ve been working from home more than I’ve been going to Princeton/IAS. My goal is to change that soon and I actually had a wonderful day at IAS today. I’d like to go tomorrow but I have a work meeting at the least convenient time one can imagine; there’s also no topology seminar at the University tomorrow, so I suppose I’ll be staying in and working again. No harm no foul, I suppose.

So what have I been working on? Well:

Universal Circles for Depth-One Foliations of 3-Manifolds. The gist here is: If you have a taut (e.g.) foliation on a 3-manifold, a theorem of Candel says we can find a metric on all the leaves so that they’re hyperbolic. Moreover, by tautness, you can lift to a foliation of the universal cover which is then a foliation whose leaves are hyperbolic discs. A ridiculously deep idea of Thurston was to look at the infinite circle boundaries of these disk leaves and maybe…glue them together? Canonically? And see if that gives insight about things?

You probably already know how this ends: It’s doable (because he’s Thurston) and it does provide deep insight about the downstairs manifold (see, e.g., the articles by Calegari & Dunfield and/or Fenley, or Calegari’s book…)

Now, let’s say we do this for certain classes of kind-of-understood-but-still-unknown-enough-to-be-interesting foliations like those of finite depth. Can we get cool manifold stuff by doing this process? I dunno, but maybe.

Homologies. My ATE was about Gabai’s work on foliating sutured manifolds, so studying sutured manifolds is something I’m still interested in. One way of doing that nowadays is with this colossal, ridiculously-powerful tool called Sutured Floer homology. So…you know…homology…but when talking with other grad students about the millions of homologies out there and about how nobody really understands what motivates discovers of them, I realized that there was a lot I needed to know before focusing on one homology foreverever. So I’m working on learning stuff about homologies.

Geometric Group Theory. Ian Agol is at IAS this year as the distinguished visitor and a lot of his work is on relationships between GGT and 3-manifolds. If you listen to any talk relating those two things, you realize there’s this whole dictionary of words and acronyms like QCERF and LERF and RAAG and Virtually Special, Residually Finite, etc. etc. I think in order to someday bridge the gap towards doing work like those guys do, I need to know what all these words mean, and what better time to figure that out than right now?! So yea…I’m doing that some, too.

Dirac Operators, Spin manifolds,…. At some point soon, I’m going to start working on hypercomplex geometry again, and part of that will be the study of Dirac operators. So far, there are lots of perspectives on those, so we’re going to try to first establish the explicit connections between them and then maybe…do some stuff? I dunno. I also have stuff on Clifford analysis / geometry I want to look at, as well as some more things involving generalized geometries. Lots here.

Topological Quantum Computing. This is a pipe dream until I’m able to feed my family and progress on my dissertation. It’s on the radar, though.

Okay, so this was an update! I’ve also been bookmarking some interesting proofs I’ve run across so I’ll know where to look when I decide to expand things here, and…yea.

Some people may read that and deduce that it’s all downhill from here. Any time I hear that phrase to describe midterm, I’m always a bit blown away. Really, it makes me wonder: Is this what downhill feels like?

Apparently, I’m the speaker at Wednesday’s Complex Analysis seminar. Abstract and other info can be found here.

Our next topology exam is scheduled for next Friday. I’m also anticipating an exam in Galois theory around the same time.

I’m on a short deadline for picking a presentation topic for my Riemannian Geometry presentation.

So, I said all that to say: (a) I still exist. (b) Life is hectic. (c) I’m not sure when I’ll get around to posting more of Hatcher 2.1, but I’ll probably be moving on to Hatcher 2.2 here in about…30 seconds. Also, (d) I really need some down-time. And a haircut. And a drink.

I’ve officially survived the first three weeks of my second year of grad school (twice, actually). Again, I know keeping count of the days is a terrible thing to do to myself, particularly when there’s been a very small amount of good to come from my weeks thus far, but at this point, I’m sort of using that countdown as some sort of badge of accomplishment. Or something.

The coming weeks are going to be very very stressful and busy and stressful. Besides my usual load of stuff (I’m enrolled in 6 classes, I have a reading class in algebraic geometry starting up on Tuesday, I’m TAing for 1 lecture and 7 labs, and I’m trying to pick advisors / plan presentations I’ll need to give some timem soon), I also thought it was a good idea (remind me why?!) to make a poster to present at FSU’s upcoming Math Fun Day. That particular endeavor shouldn’t be especially difficult, but it requires time and time, ladies and gentlemen, is precisely what I have zero of.

Daunting is the adjective that comes to mind.

Also daunting is / was / has been the thought of continuing my goal to do all the problems in Hatcher. As you may recall, I spent the first half of summer slaving to acquire the information needed for the Chapter 0 exercises, only to have my plan for Chapter 1 totality derailed by that little piece of awesome that was my Wolfram internship. Long story short: The obsessive-compulsive part of me wants to not move forward until I hash out a Chapter 1 plan, but the This will benefit me in the class I’m taking now which, subsequently, hinges on my ability to understand Chapters 2 and 3 of Hatcher part wants to press forward.

I’m pleased to announce that the second guy won out.

In particular, my Hatcher Solutions page is showing signs of progress. It didn’t take as long as I’d predicted it to take to build that framework, and due to a random, unforeseen bout of sleeplessness at 3am this morning, I had precisely the opportunity needed to seize the moment. Right now, all those are empty pages, but I’m pleased to report that I seem to have accumulated approximately six solutions; if everything goes as planned, I’ll be taking time to update by including those as soon as possible.

In the meantime, I’m going to continue to hash out what to do about this paper. And what to do about the professors I’m emailing regarding potential advisor-hood. And what to do about the fact that I severely cut my weekend work time by spending yesterday ballin’ out of control in celebration of my wife’s birth. And what to do about….

Au revoir, internet. I bid thee well.

Oh, I just remembered: I have my first exam of the semester Friday. It’s on field theory. I’m less than pleased.

So it’s been a hectic few days around these parts, in part because of things happening on the work front and in part because tomorrow is the first day back to school for me after a six week hiatus. It’s bittersweet, really.

By and large, the learning part of school makes me happy; I guess that’s a given since it’s a career thing for me, now. Tethered to that aspect are the things that are less-pleasant, among which are miscellaneous other duties, etc. I’ll be taking one class which, for all intents and purposes, seems like it’s going to be amazing; I’ll also be spending around 8 hours per week doing TA duties, and trying to split the remainder of my time between continuing the work I’ve been doing throughout the summer, balancing work-at-home things, and seeing about an internship that may be beginning soon.

I spent today being mostly idle on the math front. My plan was to have a carryover of yesterday’s supposed Algebra day since yesterday was spent mostly idle on the math front, as well. Today consisted of lots of not feeling well, running errands, and sleeping randomly. After all that subsided, I tried to work some of the exercises in Eisenbud and Harris only to be re-re-re-reminded of how important it’s going to be for me to get a good book that incorporates category theoretic ideas into some kinds of examples so I can see how to use ideas instead of just read them.

Seriously, though: I’ve read the handful of equivalent definitions of direct limits about 300,000 times, and I’ve scoured the internet to see how people respond to other people asking how to compute them, and still: I have no idea what I’m actually trying to do. I’m not sure how many times someone has to read and reread the same four pages on sheaf theory before something clicks, but I’m starting to grow anxious.

I just wanted to drop in and update here. I haven’t been posting much in the last day or two, but not because I haven’t been workin’ it!

Here’s what’s been going on.

Wednesday, I stayed home and had a Clifford Analysis day. I read a solid three or four pages of my professor’s paper before calling it a day.

Because I felt like I hadn’t done enough on the Clifford front, I went to my office Thursday armed with new writing supplies and spent a solid few hours verifying the claims made in the aforementioned three or four pages I’d read. That was a good feeling.

Friday was (differential) geometry day, and I started the day working some “trivial” problems from Spivak’s little book. In the middle of the day, I had a phone interview with Pearson for a potential part-time job; that interview went well and I’m moving on to the second stage of the employment process. I spent some more time in Spivak’s little book before spending the remainder of my evening working problems from Volume One of Spivak’s magnum opus. Those problems are also “elementary” but they’re a bit harder. The challenge was good.

Today is supposed to be algebra day. Because we only recently were in a position to remedy some previously-existing financial woes, however, we spent most of the day split between running errands and spending time out and about with our son. I did take both Eisenbud/Harris and Perrin with me, along with my trusted G2 and Composition Book; very little progress was made, however.

I’m actually about to dip out for the evening here in a few minutes, but depending on how much energy I have tonight, I might buckle in and try to figure out some of this sheaf theory stuff. If I had a fourth Algebraic Geometry Observation published, it would be that transferring between theory and problems which apply said theory is very VERY difficult.

I was in my fourth semester (a spring semester) as a master’s student: I had passed my two mandatory abstract algebra classes my first two semesters there and had passed my comprehensive exams during the Fall (my third semester). As was custom, then, I spent my third and fourth semesters taking random “advanced topics” courses aimed at potential doctoral students, and one of the sequences I took was the algebra sequence.

My first semester doctoral-level (or 7000-level as was colloquial there) algebra class was over the classification of finite simple groups and was by far the most difficult class I’d ever taken at the time. Apparently, being a student who doesn’t remotely have the sufficient background and being in a class run by a professor who has unimaginably-greater background – who teaches as if the audience consists of peers – makes for a difficult time. I squeaked out an A.

In the second semester of 7000-algebra, however, things were far less directed. Long story short, it was a potpurri of material, some from algebraic topology, some from homological algebra, and some – about 1/3 of the course, I’d say – from category theory. That was my very first exposure to an area I didn’t otherwise know existed and I remember thinking, This is the most abstract thing that’s ever been devised, and also, There’s no way this will ever be far-reaching outside the realm of mathematics.

I’ve since realized that the first assertion isn’t really true – unsurprisingly since my exposure to other areas has increased drastically since leaving there – but apparently, the second one isn’t either. To be more precise, I stumbled upon this article online which describes a number of non-math areas that have been benefiting – and will continue to benefit – from the use of category theoretic ideas.

It’s really quite amazing to see, but in and of itself is unsurprising given the fact that category theory itself was devised to provide unity among the wide variety of subdisciplines of mathematics. As a pure mathematician, I always tried to find a balance between being interested in too broad a range of topics and being too narrow with my scope; the spread of category theory invites us all to analyze that aspect of ourselves. To borrow a quote from David Spivak’s exposition (available on the arXiv),:

It is often useful to focus ones study by viewing an individual thing, or a group of things, as though it exists in isolation. However, the ability to rigorously change our point of view, seeing our object of study in a diﬀerent context, often yields unexpected insights. Moreover this ability to change perspective is indispensable for eﬀectively communicating with and learning from others. It is the relationships between things, rather than the things in and by themselves, that are responsible for generating the rich variety of phenomena we observe in the physical, informational, and mathematical worlds.

"A good stock of examples, as large as possible, is indispensable for a thorough understanding of any concept, and when I want to learn something new, I make it my first job to build one." - Paul Halmos